INVERSE PROBLEMS FOR FRACTIONAL EQUATIONS WITH A MINIMAL NUMBER OF MEASUREMENTS

In this paper, we study several inverse problems associated with a fractional differential equation of the following form: (-Delta)(s)u(x)+(N)& sum;(k=0 )a((k))(x)[u(x)](k )= 0, 0< s < 1, N is an element of N boolean OR {0}boolean OR {infinity}, which is given in a bounded domain Omega subset of R-n, n >= 1. For any finite N, we show that a((k))(x), k = 0, 1, & mldr;, N, can be uniquely determined by N+1 different pairs of Cauchy data in Omega(e): = R-n\Omega<overline>. If N = infinity, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely s = 1, even for the simplest case when N = 0, a fortiori N >= 1. The nonlocality plays a key role in establishing the uniqueness result, and we do not utilize any linearization techniques. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.